As for the first question, see the OEIS entry of this sequence. At least this is interesting because such a complex sum and such a simple algorithm give the same result.

Please see my new (2024-11-10) similar question.

@PavelGubkin, thank you for comment! There are typos: it should be $t_{1,i}:=\nu_{1,i}, t_{2,i}:=\nu_{2,i}$. Also $A+=B$ is equivalent to $A:=A+B$. However, when we apply this operator inside the sum, it still affects the terms. The difference is that the terms are not incremented sequentially (as if it were in a cycle), but simultaneously. In other words, we can say that we reserve vector $s_1$, then apply $s_1 := \nu_1$ at each step of the cycle by $i$ and inside the sum it is $\nu_{1,j}:=\nu_{1,j} + s_{1,j-1}$ (same for $\nu_2$).

@MaxAlekseyev, thank you for comment! In this question, there are two different complicated pairs for $\gcd$ instead of one simple pair for $\gcd$ in the question you mentioned.